Algorithmic identification of probabilities is hard

Abstract

Reading more and more bits from an infinite binary sequence that is random for a Bernoulli measure with parameter p, we can get better and better approximations of p using the strong law of large numbers. In this paper, we study a similar situation from the viewpoint of inductive inference. Assume that p is a computable real, and we have to eventually guess the program that computes p. We show that this cannot be done computably, and extend this result to more general computable distributions. We also provide a weak positive result showing that looking at a sequence X generated according to some computable probability measure, we can guess a sequence of algorithms that, starting from some point, compute a measure that makes X Martin-Löf random.